The Bethe Ansatz in the Heisenberg model and the Square Ice model
Summary
Two physical models are discussed. We start out with the Heisenberg spin chain, which consists of a number of atoms in a circular con?guration, each with spin up or down. We set up a Hamiltonian of this system considering only nearest-neighbour interactions, and determine the energy of the spin chain. The mathematical description of the state in which these atoms will be, i.e. the wave functions of these atoms, satis?es a certain template, which is called the Bethe Ansatz. Using this Ansatz, we can solve the eigenvalue equation of the Hamiltonian and thus calculate the energy of the system. What follows is a discussion on a model for water ice, of which we want to determine the entropy at absolute zero. This leads to a set of equations that is very similar to that encountered in the Heisenberg model, and it turns out that the Bethe Ansatz is useful in this model as well: it enables us to explicitly calculate the entropy of an infinitely big system.